## Monday, July 3, 2017

### On Triple Ratios: 6 of 6

Complex Multiplications

Given an arithmetic of dual numbers x+ey, and a conjugation # that turns e into –e, then consider this operation:
x*y   =  ( x*y + (#x)*y + x*(#y) – (#x)*(#y) ) / 2
-         where * is dual-arithmetic multiplication: e*e = 1.
Then these laws follow:
* is commutative
If r is real, then x*(r*y)  = r *(x*y)
1*1 = 1
e*e = -1
From which follows:
* is isomorphic to complex multiplication.

Here are three complex multiplications of triple ratios:

x*1y   =  ( x*y +1 (1#x)*y +1 x*(1#y) –1 (1#x)*(1#y) ) / 21
x*2y   =  ( x*y +2 (2#x)*y +2 x*(2#y) –2 (2#x)*(2#y) ) / 22
x*3y   =  ( x*y +3 (3#x)*y +3 x*(3#y) –3 (3#x)*(3#y) ) / 23
-         where 21  =  (1;2;2)    ;   22  =  (2;1;2)   ;   23  =  (2;2;1)

Exercise for the ambitious student:

Find formulas for x*1y in terms of x1, x2, x3, y1, y2, and y3; and likewise for x*2y  and x*3y.

Open Questions

How do the three arithmetics relate, other than by S3 conjugation? Are there identities involving products of additions, reductions, or complex multiplications?

Define, for any triple ratios A = (a1, a2, a3)  and B = (b1, b2, b3) , the two functions M and m:
M(A,B)      =       (A+1B)*(A+2B)*(A+3B) / (A*B)
=   1  /  ( a2b3 + a3b2  ;  a1b3 + a3b1 ;  a1b2 + a2b1 )
m(A,B)       =       (A[+]1B)*(A[+]2B)*(A[+]3B) / (A*B)
=   (a1b1(a2b3+a3b2) ; a2b2(a1b3+a3b1) ; a3b3(a1b2+a2b1))

Since (x[+]ay)*(x+ay)   = x*y, it follows that:
M(A,B)*m(a,b)             =       A*B
m(a,b)                  =       A*B / M(A,B)
1/m(A,B)             =       M(1/A, 1/B)
1/M(A,B)            =       m(1/A, 1/B)

For any triple ratios A, B and C:
M(A,A)                =       A
m(A,A)                =       A
A * M(B,C)         =       M(A*B, A*C)
A * m(B,C)         =       m(A*B, A*C)
A / M(B,C) =       m(A/B, A/C)
A / m(B,C) =       M(A/B, A/C)
And for any index a:
a# M(A,B)           =       M(a#A, a#B)
a# m(A,B)           =       m(a#A, a#B)
M and m are symmetric under S3, like times and divide.
How do M and m relate to the three additions?